Prove that the medians of the triangle \(ABC\) intersect at one point and that point divides the medians in a ratio of \(2: 1\), counting from the vertex.
A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.
A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).
Fill an ordinary chessboard with the tiles shown in the figure.
A car registration number consists of three letters of the Russian alphabet (that is, 30 letters are used) and three digits: first we have a letter, then three digits followed by two more letters. How many different car registration numbers are there?
An endless board is painted in three colours (each cell is painted in one of the colours). Prove that there are four cells of the same colour, located at the vertices of the rectangle with sides parallel to the side of one cell.
There are 17 carriages in a passenger train. How many ways can you arrange 17 conductors around the carriages if one conductor has to be in each carriage?
How many ways can you choose four people for four different positions, if there are nine candidates for these positions?
Out of two mathematicians and ten economists, it is necessary to form a committee made up of eight people. In how many ways can a committee be formed if it has to include at least one mathematician?
There are \(n\) points on the plane. How many lines are there with endpoints at these points?